Equilibrium points of non-atomic games

On Equilibrium in Pure Strategies in Games with Many Players

“…14 Also, both assume …nite sets of pure strategies. In these respects our Theorem 2 extends that due to Rashid (1983) and Kalai (2002). 15 1 3 Indeed, Kalai demonstrates that not only can a Nash equilibrium be puri…ed but when a Nash equilibrium is played almost any realized set of strategy pro…les must be an approximate Nash equilibrium.…”

On the existence of pure-strategy equilibria in large games

“…These include those of Khan and Sun [10] and Podczeck [15], which consider a richer measure space of players, that of Mas-Colell [11], where the equilibrium notion is formalized as a distribution, and that of Rashid [16], which considers approximate equilibria in games with a large but finite set of players. 1 Clearly, an existence theorem that allows for general compact action spaces also allows for finite action spaces.…”

On the purification of Nash equilibria of large games

“…This class of games is interesting because all Nash equilibria of any such game can be purified, whenever the set of players is described by a non-atomic measure space (see Schmeidler [6]). Salim Rashid [5] stated an asymptotic version of Schmeidler's theorem, according to which all Nash equilibria of sufficiently large games of the same class can be approximately purified. Asymptotic results are important because, after all, real-world games have finitely many players.…”